\(\int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} (30 x-6 x^2+x^3+(120-24 x) \log (3))+(-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3) \log (-4+e^{\frac {2 x+4 \log (3)}{x^2}})}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx\) [3365]

Optimal result

Integrand size = 112, antiderivative size = 26 \[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=(-5+x) \left (\frac {1}{3}+\log \left (-4+e^{\frac {2+\frac {4 \log (3)}{x}}{x}}\right )\right ) \]

[Out]

(1/3+ln(exp((2+4*ln(3)/x)/x)-4))*(-5+x)

Rubi [F]

\[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=\int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx \]

[In]

Int[(-4*x^3 + E^((2*x + 4*Log[3])/x^2)*(30*x - 6*x^2 + x^3 + (120 - 24*x)*Log[3]) + (-12*x^3 + 3*E^((2*x + 4*L
og[3])/x^2)*x^3)*Log[-4 + E^((2*x + 4*Log[3])/x^2)])/(-12*x^3 + 3*E^((2*x + 4*Log[3])/x^2)*x^3),x]

[Out]

x/3 - 4*Log[-4 + 9^(2/x^2)*E^(2/x)]*Defer[Int][(-4 + 81^x^(-2)*E^(2/x))^(-1), x] + (Log[-4 + 9^(2/x^2)*E^(2/x)
]*Defer[Int][(3^(1 + 4/x^2)*E^(2/x))/(-4 + 81^x^(-2)*E^(2/x)), x])/3 + (40*Log[3]*Defer[Int][(3^(1 + 4/x^2)*E^
(2/x))/((-4 + 81^x^(-2)*E^(2/x))*x^3), x])/3 - (2*Defer[Int][(3^(1 + 4/x^2)*E^(2/x))/((-4 + 81^x^(-2)*E^(2/x))
*x), x])/3 - 2*Log[81]*Defer[Int][(E^((2*(x + Log[9]))/x^2)*Defer[Int][(-4 + 81^x^(-2)*E^(2/x))^(-1), x])/((-2
+ 9^x^(-2)*E^x^(-1))*x^3), x] + 2*Log[81]*Defer[Int][(E^((2*(x + Log[9]))/x^2)*Defer[Int][(-4 + 81^x^(-2)*E^(
2/x))^(-1), x])/((2 + 9^x^(-2)*E^x^(-1))*x^3), x] - 2*Defer[Int][(E^((2*(x + Log[9]))/x^2)*Defer[Int][(-4 + 81
^x^(-2)*E^(2/x))^(-1), x])/((-2 + 9^x^(-2)*E^x^(-1))*x^2), x] + 2*Defer[Int][(E^((2*(x + Log[9]))/x^2)*Defer[I
nt][(-4 + 81^x^(-2)*E^(2/x))^(-1), x])/((2 + 9^x^(-2)*E^x^(-1))*x^2), x] + (Log[81]*Defer[Int][(E^((2*(x + Log
[9]))/x^2)*Defer[Int][(3^(1 + 4/x^2)*E^(2/x))/(-4 + 81^x^(-2)*E^(2/x)), x])/((-2 + 9^x^(-2)*E^x^(-1))*x^3), x]
)/6 - (Log[81]*Defer[Int][(E^((2*(x + Log[9]))/x^2)*Defer[Int][(3^(1 + 4/x^2)*E^(2/x))/(-4 + 81^x^(-2)*E^(2/x)
), x])/((2 + 9^x^(-2)*E^x^(-1))*x^3), x])/6 + Defer[Int][(E^((2*(x + Log[9]))/x^2)*Defer[Int][(3^(1 + 4/x^2)*E
^(2/x))/(-4 + 81^x^(-2)*E^(2/x)), x])/((-2 + 9^x^(-2)*E^x^(-1))*x^2), x]/6 - Defer[Int][(E^((2*(x + Log[9]))/x
^2)*Defer[Int][(3^(1 + 4/x^2)*E^(2/x))/(-4 + 81^x^(-2)*E^(2/x)), x])/((2 + 9^x^(-2)*E^x^(-1))*x^2), x]/6 - 2*(
5 - Log[81])*Defer[Subst][Defer[Int][E^(2*x + 4*x^2*Log[3])/(-4 + 81^x^2*E^(2*x)), x], x, x^(-1)]

Rubi steps \begin{align*} \text {integral}& = \int \frac {4 x^3-e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )-\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{3 \left (4-81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx \\ & = \frac {1}{3} \int \frac {4 x^3-e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )-\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{\left (4-81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx \\ & = \frac {1}{3} \int \left (-\frac {4 \left (1+3 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right )}{-4+81^{\frac {1}{x^2}} e^{2/x}}+\frac {81^{\frac {1}{x^2}} e^{2/x} \left (6 x^2-x^3-30 x \left (1-\frac {4 \log (3)}{5}\right )-120 \log (3)-3 x^3 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right )}{\left (4-81^{\frac {1}{x^2}} e^{2/x}\right ) x^3}\right ) \, dx \\ & = \frac {1}{3} \int \frac {81^{\frac {1}{x^2}} e^{2/x} \left (6 x^2-x^3-30 x \left (1-\frac {4 \log (3)}{5}\right )-120 \log (3)-3 x^3 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right )}{\left (4-81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-\frac {4}{3} \int \frac {1+3 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ & = \frac {1}{3} \int \left (\frac {81^{\frac {1}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}}-\frac {2\ 3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x}+\frac {40\ 3^{1+\frac {4}{x^2}} e^{2/x} \log (3)}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3}-\frac {2\ 3^{1+\frac {4}{x^2}} e^{2/x} (-5+\log (81))}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^2}+\frac {3^{1+\frac {4}{x^2}} e^{2/x} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )}{-4+81^{\frac {1}{x^2}} e^{2/x}}\right ) \, dx-\frac {4}{3} \int \left (\frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}}+\frac {3 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )}{-4+81^{\frac {1}{x^2}} e^{2/x}}\right ) \, dx \\ & = \frac {1}{3} \int \frac {81^{\frac {1}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx+\frac {1}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-\frac {4}{3} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-4 \int \frac {\log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx+\frac {1}{3} (2 (5-\log (81))) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^2} \, dx \\ & = \frac {1}{3} \int \left (1+\frac {4}{-4+81^{\frac {1}{x^2}} e^{2/x}}\right ) \, dx-\frac {1}{3} \int \frac {2 e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (4-e^{\frac {2 (x+\log (9))}{x^2}}\right ) x^3} \, dx-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-\frac {4}{3} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx+4 \int \frac {2 e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (4-e^{\frac {2 (x+\log (9))}{x^2}}\right ) x^3} \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-\frac {1}{3} (2 (5-\log (81))) \text {Subst}\left (\int \frac {3^{1+4 x^2} e^{2 x}}{-4+81^{x^2} e^{2 x}} \, dx,x,\frac {1}{x}\right )+\frac {1}{3} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right ) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\left (4 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right ) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ & = \frac {x}{3}-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-\frac {2}{3} \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (4-e^{\frac {2 (x+\log (9))}{x^2}}\right ) x^3} \, dx+8 \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (4-e^{\frac {2 (x+\log (9))}{x^2}}\right ) x^3} \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-\frac {1}{3} (2 (5-\log (81))) \text {Subst}\left (\int \frac {3 e^{2 x+4 x^2 \log (3)}}{-4+81^{x^2} e^{2 x}} \, dx,x,\frac {1}{x}\right )+\frac {1}{3} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right ) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\left (4 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right ) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ & = \frac {x}{3}-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-\frac {2}{3} \int \left (-\frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{4 \left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}+\frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{4 \left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}\right ) \, dx+8 \int \left (-\frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{4 \left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}+\frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{4 \left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}\right ) \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-(2 (5-\log (81))) \text {Subst}\left (\int \frac {e^{2 x+4 x^2 \log (3)}}{-4+81^{x^2} e^{2 x}} \, dx,x,\frac {1}{x}\right )+\frac {1}{3} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right ) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\left (4 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right ) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ & = \frac {x}{3}+\frac {1}{6} \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx-\frac {1}{6} \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-2 \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx+2 \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} (x+\log (81)) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-(2 (5-\log (81))) \text {Subst}\left (\int \frac {e^{2 x+4 x^2 \log (3)}}{-4+81^{x^2} e^{2 x}} \, dx,x,\frac {1}{x}\right )+\frac {1}{3} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right ) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\left (4 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right ) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ & = \frac {x}{3}+\frac {1}{6} \int \left (\frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2}+\frac {e^{\frac {2 (x+\log (9))}{x^2}} \log (81) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}\right ) \, dx-\frac {1}{6} \int \left (\frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2}+\frac {e^{\frac {2 (x+\log (9))}{x^2}} \log (81) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}\right ) \, dx-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-2 \int \left (\frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2}+\frac {e^{\frac {2 (x+\log (9))}{x^2}} \log (81) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}\right ) \, dx+2 \int \left (\frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2}+\frac {e^{\frac {2 (x+\log (9))}{x^2}} \log (81) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3}\right ) \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-(2 (5-\log (81))) \text {Subst}\left (\int \frac {e^{2 x+4 x^2 \log (3)}}{-4+81^{x^2} e^{2 x}} \, dx,x,\frac {1}{x}\right )+\frac {1}{3} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right ) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\left (4 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right ) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ & = \frac {x}{3}+\frac {1}{6} \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2} \, dx-\frac {1}{6} \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2} \, dx-\frac {2}{3} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x} \, dx-2 \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2} \, dx+2 \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^2} \, dx+\frac {1}{3} (40 \log (3)) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{\left (-4+81^{\frac {1}{x^2}} e^{2/x}\right ) x^3} \, dx-(2 (5-\log (81))) \text {Subst}\left (\int \frac {e^{2 x+4 x^2 \log (3)}}{-4+81^{x^2} e^{2 x}} \, dx,x,\frac {1}{x}\right )+\frac {1}{6} \log (81) \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx-\frac {1}{6} \log (81) \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx-(2 \log (81)) \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (-2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx+(2 \log (81)) \int \frac {e^{\frac {2 (x+\log (9))}{x^2}} \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx}{\left (2+9^{\frac {1}{x^2}} e^{\frac {1}{x}}\right ) x^3} \, dx+\frac {1}{3} \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right ) \int \frac {3^{1+\frac {4}{x^2}} e^{2/x}}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx-\left (4 \log \left (-4+e^{\frac {2 (x+\log (9))}{x^2}}\right )\right ) \int \frac {1}{-4+81^{\frac {1}{x^2}} e^{2/x}} \, dx \\ \end{align*}

Mathematica [F]

\[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=\int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx \]

[In]

Integrate[(-4*x^3 + E^((2*x + 4*Log[3])/x^2)*(30*x - 6*x^2 + x^3 + (120 - 24*x)*Log[3]) + (-12*x^3 + 3*E^((2*x
+ 4*Log[3])/x^2)*x^3)*Log[-4 + E^((2*x + 4*Log[3])/x^2)])/(-12*x^3 + 3*E^((2*x + 4*Log[3])/x^2)*x^3),x]

[Out]

Integrate[(-4*x^3 + E^((2*x + 4*Log[3])/x^2)*(30*x - 6*x^2 + x^3 + (120 - 24*x)*Log[3]) + (-12*x^3 + 3*E^((2*x
+ 4*Log[3])/x^2)*x^3)*Log[-4 + E^((2*x + 4*Log[3])/x^2)])/(-12*x^3 + 3*E^((2*x + 4*Log[3])/x^2)*x^3), x]

Maple [A] (verified)

Time = 6.85 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50

method	result	size
risch	\(x \ln \left (81^{\frac {1}{x^{2}}} {\mathrm e}^{\frac {2}{x}}-4\right )+\frac {x}{3}-5 \ln \left (81^{\frac {1}{x^{2}}} {\mathrm e}^{\frac {2}{x}}-4\right )\)	\(39\)
parallelrisch	\(x \ln \left ({\mathrm e}^{\frac {4 \ln \left (3\right )+2 x}{x^{2}}}-4\right )+\frac {x}{3}-5 \ln \left ({\mathrm e}^{\frac {4 \ln \left (3\right )+2 x}{x^{2}}}-4\right )\)	\(39\)
norman	\(\frac {x^{3} \ln \left ({\mathrm e}^{\frac {4 \ln \left (3\right )+2 x}{x^{2}}}-4\right )-5 x^{2} \ln \left ({\mathrm e}^{\frac {4 \ln \left (3\right )+2 x}{x^{2}}}-4\right )+\frac {x^{3}}{3}}{x^{2}}\)	\(52\)

[In]

int(((3*x^3*exp((4*ln(3)+2*x)/x^2)-12*x^3)*ln(exp((4*ln(3)+2*x)/x^2)-4)+((-24*x+120)*ln(3)+x^3-6*x^2+30*x)*exp
((4*ln(3)+2*x)/x^2)-4*x^3)/(3*x^3*exp((4*ln(3)+2*x)/x^2)-12*x^3),x,method=_RETURNVERBOSE)

[Out]

x*ln(81^(1/x^2)*exp(2/x)-4)+1/3*x-5*ln(81^(1/x^2)*exp(2/x)-4)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88 \[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx={\left (x - 5\right )} \log \left (e^{\left (\frac {2 \, {\left (x + 2 \, \log \left (3\right )\right )}}{x^{2}}\right )} - 4\right ) + \frac {1}{3} \, x \]

[In]

integrate(((3*x^3*exp((4*log(3)+2*x)/x^2)-12*x^3)*log(exp((4*log(3)+2*x)/x^2)-4)+((-24*x+120)*log(3)+x^3-6*x^2
+30*x)*exp((4*log(3)+2*x)/x^2)-4*x^3)/(3*x^3*exp((4*log(3)+2*x)/x^2)-12*x^3),x, algorithm="fricas")

[Out]

(x - 5)*log(e^(2*(x + 2*log(3))/x^2) - 4) + 1/3*x

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=x \log {\left (e^{\frac {2 x + 4 \log {\left (3 \right )}}{x^{2}}} - 4 \right )} + \frac {x}{3} - 5 \log {\left (e^{\frac {2 x + 4 \log {\left (3 \right )}}{x^{2}}} - 4 \right )} \]

[In]

integrate(((3*x**3*exp((4*ln(3)+2*x)/x**2)-12*x**3)*ln(exp((4*ln(3)+2*x)/x**2)-4)+((-24*x+120)*ln(3)+x**3-6*x*
*2+30*x)*exp((4*ln(3)+2*x)/x**2)-4*x**3)/(3*x**3*exp((4*ln(3)+2*x)/x**2)-12*x**3),x)

[Out]

x*log(exp((2*x + 4*log(3))/x**2) - 4) + x/3 - 5*log(exp((2*x + 4*log(3))/x**2) - 4)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (27) = 54\).

Time = 0.32 (sec) , antiderivative size = 99, normalized size of antiderivative = 3.81 \[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=\frac {3 \, x^{2} \log \left (e^{\left (\frac {1}{x} + \frac {2 \, \log \left (3\right )}{x^{2}}\right )} + 2\right ) + 3 \, x^{2} \log \left (e^{\left (\frac {1}{x} + \frac {2 \, \log \left (3\right )}{x^{2}}\right )} - 2\right ) + x^{2} - 30}{3 \, x} - 5 \, \log \left ({\left (e^{\left (\frac {1}{x} + \frac {2 \, \log \left (3\right )}{x^{2}}\right )} + 2\right )} e^{\left (-\frac {1}{x}\right )}\right ) - 5 \, \log \left ({\left (e^{\left (\frac {1}{x} + \frac {2 \, \log \left (3\right )}{x^{2}}\right )} - 2\right )} e^{\left (-\frac {1}{x}\right )}\right ) \]

[In]

integrate(((3*x^3*exp((4*log(3)+2*x)/x^2)-12*x^3)*log(exp((4*log(3)+2*x)/x^2)-4)+((-24*x+120)*log(3)+x^3-6*x^2
+30*x)*exp((4*log(3)+2*x)/x^2)-4*x^3)/(3*x^3*exp((4*log(3)+2*x)/x^2)-12*x^3),x, algorithm="maxima")

[Out]

1/3*(3*x^2*log(e^(1/x + 2*log(3)/x^2) + 2) + 3*x^2*log(e^(1/x + 2*log(3)/x^2) - 2) + x^2 - 30)/x - 5*log((e^(1
/x + 2*log(3)/x^2) + 2)*e^(-1/x)) - 5*log((e^(1/x + 2*log(3)/x^2) - 2)*e^(-1/x))

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.46 \[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=x \log \left (e^{\left (\frac {2 \, {\left (x + 2 \, \log \left (3\right )\right )}}{x^{2}}\right )} - 4\right ) + \frac {1}{3} \, x - 5 \, \log \left (e^{\left (\frac {2 \, {\left (x + 2 \, \log \left (3\right )\right )}}{x^{2}}\right )} - 4\right ) \]

[In]

integrate(((3*x^3*exp((4*log(3)+2*x)/x^2)-12*x^3)*log(exp((4*log(3)+2*x)/x^2)-4)+((-24*x+120)*log(3)+x^3-6*x^2
+30*x)*exp((4*log(3)+2*x)/x^2)-4*x^3)/(3*x^3*exp((4*log(3)+2*x)/x^2)-12*x^3),x, algorithm="giac")

[Out]

x*log(e^(2*(x + 2*log(3))/x^2) - 4) + 1/3*x - 5*log(e^(2*(x + 2*log(3))/x^2) - 4)

Mupad [B] (verification not implemented)

Time = 9.22 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int \frac {-4 x^3+e^{\frac {2 x+4 \log (3)}{x^2}} \left (30 x-6 x^2+x^3+(120-24 x) \log (3)\right )+\left (-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3\right ) \log \left (-4+e^{\frac {2 x+4 \log (3)}{x^2}}\right )}{-12 x^3+3 e^{\frac {2 x+4 \log (3)}{x^2}} x^3} \, dx=\frac {x}{3}-5\,\ln \left (3^{\frac {4}{x^2}}\,{\mathrm {e}}^{2/x}-4\right )+x\,\ln \left (3^{\frac {4}{x^2}}\,{\mathrm {e}}^{2/x}-4\right ) \]

[In]

int((log(exp((2*x + 4*log(3))/x^2) - 4)*(3*x^3*exp((2*x + 4*log(3))/x^2) - 12*x^3) + exp((2*x + 4*log(3))/x^2)
*(30*x - log(3)*(24*x - 120) - 6*x^2 + x^3) - 4*x^3)/(3*x^3*exp((2*x + 4*log(3))/x^2) - 12*x^3),x)

[Out]

x/3 - 5*log(3^(4/x^2)*exp(2/x) - 4) + x*log(3^(4/x^2)*exp(2/x) - 4)